What is the Centroid of a Cylindrical Cone?

[zandria]

Homework Statement

Determine the centroid of volume for a right circular cone with base diameter of 100mm and an altitude of 200mm.

Homework Equations

I know that if the my xy-plane is parallel to the base of the cylindrical cone then the x and y coordinates of the centroid must be zero and therefore I only need to find the z coordinate of the centroid.

The equation I am using is

$$z_c = \frac{1}{M} \int_{body} z dm$$

where M is the total mass and $$dm = \rho dV$$

The Attempt at a Solution

I am trying to use cylindrical coordinates but I think my limits of integration are incorrect. I have tried to solve the integral above with the following limits.

$$0<\theta<2\pi$$

$$0<r<50$$

$$0<z<(200-r/4)$$

I'm not sure if the limits for the z coordinate is correct. Am I on the right path?

 

[LowlyPion]

Well, I wouldn't worry with polar coordinates, because you are dealing with basically a stack of disks aren't you?

They are each have a weight of ρ*πr²

Exploit then the fact that r is a function of z, and your integral should be pretty straight forward shouldn't it?

 

[zandria]

Thank you. I was essentially doing the right thing on my first try before I changed everything, but I made an algebra mistake when trying to use cylindrical coordinates. Thanks for the short cut ... less room for stupid mistakes.